We attempt to solve the Einstein equations for string dust and null flowing
radiation for the general axially symmetric metric, which we believe is being
done for the first time. We obtain the string-dust and radiating
generalizations of the Kerr and the NUT solutions. There also occurs an
interesting case of radiating string-dust which arises from string-dust
generalization of Vaidya's solution of a radiating star.Comment: 15 pages, TeX version. tex error correcte

Beesham, A.

Dadhich, N.

Patel, L. K.

English

arXiv

We attempt to solve the Einstein equations for string dust and null flowing radiation for the general axially symmetric metric, which we believe is being done for the first time. We obtain the string-dust and radiating generalizations of the Kerr and the NUT solutions. There also occurs an interesting case of radiating string-dust which arises from string-dust generalization of Vaidya's solution of a radiating star

Patel, L K

Dadhich, N K

Beesham, A

CERN Document Server

/June 1995
STRING-DUST DISTRIBUTIONS WITH
THE KERR-NUT SYMMETRY
L.K. Patel
1
, N. Dadhich
2
and A. Beesham
1
.
1. Department of Applied Mathematics, University of Zululand, Private Bag
X1001, Kwa-Dlangezwa 3886, South Africa.
2. Inter-University Centre for Astronomy & Astrophysics, Post Bag 4,
Ganeshkhind, Pune - 411 007, India.
Abstract
We attempt to solve the Einstein equations for string dust and null owing
radiation for the general axially symmetric metric, which we believe is being done
for the rst time. We obtain the string-dust and radiating generalizations of the
Kerr and the NUT solutions. There also occurs an interesting case of radiating
string-dust which arises from string-dust generalization of Vaidya's solution of a
radiating star.
Key words : String dust, Radiating Kerr and NUT, Null Radiation, Radiating
String dust.
PACS No : 04.20Jb, 0.4.20 Cv,
1
Two empty space solutions of Einstein's equations admitting twisting, shear-
free and geodetic null rays are well-known in the literature. They are the Kerr
solution [1] and the so called Taub-NUT solution [2]. They are remarkable solutions,
the former has important astrophysical applications while the latter is interesting
on the formal grounds. Vaidya [3] has expressed the usual Kerr metric in the form
ds
2
= 2(du+ ksin
2
d)dt  (r
2
+ k
2
cos
2
)(d
2
+ sin
2
d
2
)
 

1 +
2mr
r
2
+ k
2
cos
2


(du+ ksin
2
d)
2
(1:1)
where u = t   r and m and k are arbitrary constants indicating mass and rotation
parameters of the Kerr particle. On the other hand the NUT solution is given by
ds
2
= 2(du  2bcosd)dr   (r
2
+ b
2
)(d
2
+ sin
2
d
2
)
 

1 +
2mr   2b
2
r
2
+ b
2

(du  2bcosd)
2
(1:2)
where constants m and b are the parameters of the NUT source. It can be easily
veried that the Schwarzschild solution will result when k = 0 in (1.1) or b = 0 in
(1.2). The Kerr solution is asymptotically at while the NUT is not.
Further Vaidya et al [4] have synthesized the two into one by writing
ds
2
= 2(du+ g sind)dx  M
2
(d
2
+ sin
2
d
2
)   2L(du+ g sind)
2
(1:3)
with g = g(), M and L are functions of u; x and . The coordinate x = t for the
Kerr and x = r for the NUT, and u = t   r always. In this framework they have
obtained some radiating solutions; i.e. Kerr or NUT source with outowing null
radiation.
In this paper we wish to nd for the metric (1.3) solutions of the equation
R
ik
=  8K(u
i
u
k
  g
ik
  w
i
w
k
)   8
i

k
(1:4)
2
i i

i

i
p gy
momentum of the owing null radiation of density  while the rst corresponds to
the energy-momentum tensor
T
ik
= K(u
i
u
k
  w
i
w
k
) (1:5)
which is supposed to describe a string-dust distribution [5]. The string-dust
generalization of the Schwarzschild eld is well known and some exact solutions
have been obtained [6-8] of the equation (1.4). It is perhaps for the rst time
axially symmetric (metric (1.4)) solutions of this equation are being attempted.
Alternatively one of us [9] would like to interpret the rst term in (1.4) as
dening the Machian vacuum. Eqn.(1.5) implies R
ik
u
i
u
k
= 0 and T
0
0
= T
1
1
= K,
the rest of T
k
i
= 0. This means that the gravitational charge ( + 3p) density of
the distribution (1.5) is zero implying no inuence on free particles. Though it does
not produce gravitational force on free particles, the distribution (1.5) does produce
non-zero curvature which in spherical symmetry corresponds to constant (non-zero)
gravitational potential [9,10]. It is argued that by dening the vacuum by the rst
term in (1.4), the Schwarzschild eld can be made consistent with Mach's Principle
in the sense that homogeneous and isotropic matter distribution in the Universe
lying exterior to a spherical cavity centred at the mass point can manifest itself
by producing a constant potential. This is why it is called the equation for the
Machian vacuum [9].
The consideration of eqn. (1.4) can also be motivated by the fact that very
close to the big-bang singularity, the Universe is in highly dense state and hence
its matter content can have very unusual and exotic properties to allow for viscous
eects, heat ux, null radiation ow, string-dust etc. Some cylindrical models
3
p g p [ ]
nd the axially symmetric solutions of eqn. (1.4).
2. Field Equations
For the metric (1.3) we introduce the orthogonal tetrads as

1
= du+ gsind; 
2
=Md;

3
=Msind; 
4
= dx   L
1
and in what follows all the quantitites will be referred to the tetrad frame. The
Ricci components of the metric (1.3) are given in the Appendix I [4]. Let us note
that g

= @g=@x

; M
xu
= @
2
2M=@x@u etc. and
2f = g

+ gcot

(2:1)
and a new variable y is dened by gd = dy.
We employ the comoving coordinates to write
u
a
= (1; 0; 0; 1=2); w
a
= (1; 0; 0; 1=2); 
a
= (1; 0; 0; 0)
where the null radiation is taken to ow along the 
1
- direction.
For the metric (1.3), eqns. (1.4) imply R
23
= 0 and R
22
= R
33
and the
following system of equations.
R
44
= 0; R
42
= 0; R
43
= 0 (2:2)
R
41
= 0 (2:3)
4
R12
= 0; R
13
= 0 (2:4)
R
22
=  8K (2:5)
R
11
=  8 (2:6)
Eqns. (2.2) involve the only one function M (App. I) and their solution is given by
M
2
=
f
Y
(X
2
+ Y
2
) (2:7)
where X = X(x; u; y) and Y = Y (u; y) and they satisfy the conditions
X
x
=  1; X
y
= Y
u
; X
u
=  Y
y
: (2:8)
Next consider (2.3) which can be solved for the metric function 2L,
2L =  
Y
u
Y
X + 2G+
2FX + 2EY
X
2
+ Y
2
(2:9)
where E;F; and G are functions of u and y satisfying the relation
E =  2YG   Y Y
y
(2:10)
From (2.7) - (2.10), it follows after a lengthy algebraic manipulations,
E
u
= F
y
; E
y
=  F
u
(2:11)
Then the string density is given by (2.5),
5
8K =
1
(X
2
+ Y
2
)

2G+
Y
f
f
g
2
2
5
2
ln(Y=f)   f
y
+ 3f
Y
y
Y
+ 1g

(2:12)
where 5
2
= @
2
=@u
2
+ @
2
=@y
2
. Using the above relations, (2.6) will give the null
radiation density , the expression for which is quite lengthy and it is given in the
Appendix II.
For nding the explicit solutions we have to obtain X;Y and f from eqns.
(2.8), (2.10) and (2.11). For further investigation we shall assume f = Y .
3. The case f = Y
If f = Y , then Y becomes a function of y alone. Eqns. (2.7) and (2.8) then
give
M
2
= X
2
+ Y
2
; X = au  x; Y =  ay + b (3:1)
where a and b are constants of integration.
Eqns. (2.8), (2.10) and (2.11) will then lead to
Y 5
2
G  2aG
y
= 0 (3:2)
of which we take the particular solution,
2G = const: = c (3:3)
Then (2.11) will give
E = (a  c)Y; F = a(a   c)U +N (3:4)
where N is a constant of integration.
6
y q p y p f ,
in view of (2.1) will read as
(1   z
2
)Y
zz
  2zY
z
+ 2aY = 0; z = cos (3:5)
As seen by Vaidya et al [4], this equation admits a power series solution for
a   1=8. Writing 2a = n(n+ 1), it takes to the familiar Legendre equation,
(1  z
2
)Y
zz
  2zY
z
+ n(n + 1)Y = 0 (3:6)
which can be solved for Y . In view of gd = dy; (3.1) gives the remaining metric
potential,
g =  
1
a
Y

=
1
a
Y
z
sin

(3:7)
Thus the metric (1.3) is completely determined for the eld equations (1.4). The
string dust and the null radiation densities are then given by
8K =  

2a   c  1
X
2
+ Y
2

(3:8)
and
8 =
2a(a   c)
X
2
+ Y
2
(3:9)
It should be noted that K = 0 for 1 + c = 2a and we recover the radiating
case of Vaidya et al [4]. On the other hand  = 0 for a = 0 or a = c presenting a
string-dust spacetime.
7
When n is a positive integer, we can take the solution of (3.6) in the form
Y = AP
n
(z) +BQ
n
(z) (4:1)
where A and B are arbitrary constants, and P
n
(z) and Q
n
(z) are respectively the
Legendre and associated Legendre polynomials of order n.
The metric (1.3) is then given by
2L = c+
2(a   c)(Y
2
+ auX) + 2NX
X
2
+ Y
2
gsin =
1
a

A
dP
n
dz
+
BdQ
n
dz

sin
2

9
>
>
=
>
>
;
(4:2)
where X = au   x and Y as given in (4.1). Here x is a timelike coordinate and
hence replace x by t in X as well as in (1.3).
When 1+ c = 2a, the string-dust density vanishes and we recover the radiating
Kerr metric [4] for B = 0 while A = 0 gives the associated radiating Kerr metric
[15]. The associated Legendre function Q
n
(z) has a singularity on the axis  = 0,
so would have the associated metric. The radiation density vanishes for a = 0 or
a = c but a = 0 leads to g = 0 in view of (3.1) and (2.1) and hence a = c for the
radiation free string-dust solution. Then
8K =
1  c
X
2
+ Y
2
(4:3)
which means c  1 for K  0. Here we have X = cu   t; Y =  cy + b, the
time dependence in X is spurious can be removed by X = R = (c   1)t   cr, the
new radial coordinate. We now take B = 0 to seek the string-dust generalization
of the Kerr metric, which results when c = 1; i.e. n = 1 and from (4.1)
Y = AP
1
(z) = Acos. For the string-dust we shall hence have to take n > 1,
8
y2
( )
2
( ) p
straight way to get to the Kerr metric from the string-dust because (4.1) admits
dierent solutions in the two cases for n = 1; 2. Since n = 2 implies c = 3, from
(4.3) the string density K will be negative and so would be the case for all n > 1.
Following the standard procedure [16], we can bring the metric to the standard
Boyer-Lindquist form,
ds
2
=
2c  
c
2
dt
2
  (R
2
+ Y
2
)(9A
2
cos
2
sin
2
+ (R
2
+ Y
2
)(2c  ))
 1
dR
2
  (R
2
+ Y
2
)d
2
  (R
2
+ Y
2
+ 9
A
2
c
2
cos
2
sin
2
)sin
2
d
2
+ 6
A
c
2
(c  )cossin
2
dtd
(4:4)
where
 = c+
2mR
R
2
+ Y
2
(4:5)
Similarly one can easily obtain the metric for the string-dust generalization of
the associated Kerr metric. Clearly A is the rotation parameter and m is the mass.
A = 0 will give the Schwarzschild string-dust. The above metric apparently has
many interesting properties, such as the inherent angular velocity w =  g
03
=g
33
vanishes at both  = 0 and  = =2, which will be considered separately [17].
5. The NUT- like solutions.
We now consider the equation (3.6) for 0  n  1, a particular solution of
which is given by [4],
Y = b[1   n(n + 1)p
n
(z)] (5:1)
where p
n
(z) stands for the sum of the innite convergent series,
9
12
z
2
+
1
4!
(2  n)(3 + n)z
4
+
1
6!
(2  n)(4   n)(3 + n)(5 + n)z
6
+ : : : (5:2)
The metric functions 2L as given by (4.2) and
gsin =  2b
dp
n
dz
sin
2
 (5:3)
where X = au   r (replacing the spacelike coordinate x by r) and Y as given by
(5.1). The solution is hence given by 2L in (4.2), (5.1) and (5.3), with 2a = n(n+1).
The series (5.2) diverges as z  ! 1 and consequently the solution has a singularity
at  = 0, for all n 6= 0. The axis is thus singular. The densities are as given by
(3.8) and (3.9) with X and Y as given above. This is the string-dust generalization
of the radiaing NUT metric, discussed by Vaidya et al [4].
When a = 0, i.e. n = 0, the radiation density  = 0 (recall in the Kerr case
the appropriate conditin for  = 0 was a = c, while here it is a = 0) and in that
case the metric will describe the NUT with string-dust and is given by
ds
2
= 2(du  2bcosd)dr   (r
2
+ b
2
)(d
2
+ sin
2
d
2
)
 

c+ 2
mr   cb
2
r
2
+ b
2

(du  2bcosd)
2
(5:4)
where N = m. The string density is now given by
8K =
c+ 1
r
2
+ b
2
(5:5)
The above metric can be transformed to the BL form to read as
ds
2
= dt
2
  
 1
dr
2
  (r
2
+ b
2
)d
2
  (r
2
+ b
2
+ 4b
2
cos
2
sin
2
)
sin
2
d
2
+ 4bcossin
2
dtd
(5:6)
10
r2
+b
2
;   ,
the usual NUT metric on the other hand b = 0 implies vanishing of rotation of the
null congruence and we get the Schwarzschild string-dust.
6. Discussion
There is an important dierence between the string-dust generalizations of the
Kerr (4.4) and the NUT (5.6). From (4.4), the Kerr metric does not result when
string-dust is switched o by putting c = 1, while (5.6) yields the usual NUT metric
for c =  1. It would not be possible to match (4.4) continuously to the Kerr metric
where as (5.6) will match continuously across the boundary, r = r
0
to the NUT
metric with mass parameter m being given as
m =m+
1 + c
2r
2
0
(r
2
0
+ b
2
) (6:1)
We shall now consider an important particular case. Consider g = 0, the
spacetime (1.3) then admits a null congruence which is geodetic as well as shear
and twist free. Further take M = M(r); L = L(r; u); u = t   r, then R
44
= 0
determines M = r and eqns. R
24
= R
34
= R
12
= R
13
= 0 become identities. Eqn.
R
14
= 0 integrates to give
2L = c(u)  2
m(u)
r
(6:2)
where c(u) and m(u) are arbitrary functions of u. We have
8K =
c  1
r
2
; 8 =
2m
u
r
2
 
c
u
r
(6:3)
and the metric reads
ds
2
= 2dudt 

c(u) 
2m(u)
r

du
2
  r
2
(d
2
+ sin
2
d
2
): (6:4)
11
g g y g ,
results when c = 1. The string-density will not depend upon t for c = const: 6= 1.
The Schwarzschild string-dust will follow when m and c are constants. The
interesting case occurs when m = const: but c = c(u). This is the radiating string-
dust.
Acknowledgement : One of the authors (LKP) would like to thank the University
of Zululand, South Africa for hospitality.
12
[1] R.P. Kerr, Phys. Rev. Lett. 11 (1963) 237.
[2] E. Newman, L. Tamburino and T. Unti, J. Math. Phys. 4 (1963) 915.
[3] P.C. Vaidya, Proc. Cambridge Phi. Soc. 75 (1974) 383.
[4] P.C. Vaidya, L.K. Patel and P.V. Bhatt, Gen. Rel. Grav. 7 (1976) 701.
[5] J. Stachel, Phys. Rev. D 21 (1980) 2171.
[6] P.S. Letetier, Phys. Rev. D 20 (1979) 1274.
[7] D.R. Matraners, Gen. Rel. Grav. 20 (1988) 279.
[8] J.M. Nevin, Gen. Rel. Grav. 23 (1991) 253.
[9] N. Dadhich, Machian vacuum spacetime - to be published.
[10] |, Einstein Versus Newton: zero of gravitational potential - to be published.
[11] L.K. Patel and N. Dadhich, J. Math. Phys. 34 (1993) 1927.
[12] L.K. Patel and N. Dadhich, Astrophys. J. 401 (1992) 433.
[13] L.K. Patel and N. Dadhich, Class. Quantum Grav. 10 (1993) L85.
[14] R. Tikekar, L.K. Patel and N. Dadhich, Gen. Rel. Grav. 26 (1994) 647.
[15] P.C. Vaidya, Current Science 45 (1976) 490.
[16] R. Adler, M. Bazin and M. Schier, Introduction to General Relativity
(MacGraw-Hill 1975) pp. 254-56.
[17] N. Dadhich, - to be published.
13
pp
R
23
= 0
R
24
= (g=M)

(M
x
=M)
y
  (f=M
2
)
u

R
34
=  (g=M)

(M
x
=M)
u
+ (f=M
2
)
y

R
44
= (2=M)

M
xx
  f
2
=M
3

R
14
= (2=M)

M
xu
+ (LM
x
)
x
+ (Lf
2
=M
3
)

+L
xx
R
12
= LR
24
+ (g=M)

(L
x
+M
u
=M)
y
+ (2fL=M
2
)
u

R
13
= LR
34
+ (g=M)

  (L
x
+M
u
=M)
u
+ (2fL=M
2
)
y

R
22
= R
33
= (1=M
2
)

g
2
(M
u
=M)
u
+ g
2
(M
y
=M)
y
  1
+ 2f(My=M) + 4(f
2
L=M
2
)  (M
2
)
ux
  fL(M
2
)
x
g
x

R
11
= L
2
R
44
+ (1=M
2
)

g
2
(L
uu
+L
yy
) + 2fLy
+ 2L
u
MM
x
+ 4LMM
xu
  2L
x
MM
u
+ 2MM
uu

14
pp
R
11
=
g
2
Y
f(X
2
+ Y
2
)

5
2
G 
1
2
X

(
Y
u
Y
)
uu
+ (
Y
u
Y
)
yy

 X
u
(
Y
u
Y
)
u
  Y
u
(
Y
u
Y
)
y

+
1
(X
2
+ Y
2
)

3F
Y
u
Y
  2XG
u
+ 2Y G
y
  2F
u
  2X
2
Y
2
u
+X
2
(
Y
u
Y
)
u
 XY (
Y
u
Y
)
y
+ 2XX
uu
+ 2Y Y
uu

+
1
(X
2
+ Y
2
)
2

2E(XY
u
  Y X
u
)   Y
2
u
(X
2
+ 3Y
2
) + 2Y
2
X
2
u
  2X(X
2
+ 2Y
2
)X
u
(
Y
u
Y
) + 2GX(X
2
  Y
2
)(
Y
u
Y
)
  4GY
2
X
u

  (
Y
u
Y
)
u
15


String-Dust Distributions with the Kerr-NUT symmetry

Abstract

Similar works

Full text

Available Versions

CERN Document Server